In this paper, meromorphic function means meromorphic in the complex plane. We use the standard notations of Nevanlinna theory, which can be found in [12]. A meromorphic function a(z) is called a small function with respect to f(z) if T(r, a) = S(r, f). We say that two meromorphic functions f and g share a small function a IM (ignoring multiplicities) when f − a and g − a have the same zeros. If f − a and g − a have the same zeros with the same multiplicities, then we say that f and g share a CM (counting multiplicities).

Let f(z) be a meromorphic function. It is known that the hyper order of f(z), denoted by σ₂(f), is defined by

In 1996, R. Brück [1] posed the following conjecture.

Brück Conjecture

Let f be a non-constant entire function such that the hyper order σ₂(f) of f is not a positive integer and σ₂(f) < +∞. If f and f′ share a finite value a CM, then

where c is a nonzero constant.

In 1998, Gundersen and Yang [3] verified that the Conjecture is true when f is of finite order. In 1999, Yang [10] confirmed that the Conjecture is also true when f′ is replaced by f^(k)(k ≥ 2) and f is of finite order. In recent years, many results have been published concerning the above conjecture, see [2, 5, 7, 8, 14, 15, 16, 17, 18] etc., and Zhang [17] was the first author who considers the case when f is a meromorphic function. We need the following definition.

Definition 1.1

Let l be a non-negative integer or infinite. Denote by E_l(a, f) the set of all a-points of f where an a-point of multiplicity m is counted m times if m ≤ l and l+1 times if m > l. If E_l(a, f) = E_l(a, g), we say that f and g share (a, l).

Remark

It is easy to see that f and g share (a, l) implies that f and g share (a, p) for 0 ≤ p ≤ l. Also we note that f and g share the value a IM or CM if and only if f and g share (a, 0) or (a,∞), respectively. We also use Np(r,1f-a) to denote the counting function of the zeros of f − a where a zero of multiplicity m is counted m times if m ≤ p and p times if m > p.

Lahiri [5] improved the results of Zhang [17] by using the above definition and obtained the following Theorem:

Theorem A

Let f be a non-constant meromorphic function and k be a positive integer. If f and f^(k)share (1,2) and

for r ∞ I, where 0 < λ < 1 and I is a set of infinite linear measure, then f(k)-af-a=c for c ∞ C {0}.

Let p be a positive integer and a ∞ C ∪{∞}. We use Np)(r,1f-a) to denote the counting function of the zeros of f − a, whose multiplicities are not greater than p, N(p+1(r,1f-a) to denote the counting function of the zeros of f − a whose multiplicities are not less than p + 1, and we use N¯p)(r,1f-a) and N¯(p+1(r,1f-a) to denote their corresponding reduced counting functions (ignoring multiplicities) respectively. Define

It follows that δ_p(a, f) ≥ δ(a, f).

Let L(f) = f^(k) + a^k−1f^(k−1) + · · · + a₀f. Zhang and Yang [16] obtained the following result which improves the results of [5, 8, 14, 18].

Theorem B

Let f be a non-constant meromorphic function, k ≥ 1 and l ≥ 0 be integers. Let a(z) be a small function of f such that a(z) ≢ 0, ∞. Suppose that f −a and L(f)−a share (0, l). Then f ≡ L(f) if one of the following assumptions holds,

• (1) l ≥ 2 and3θ(∞,f)+δ2+k(0,f)+δ2(0,f)+δ(a,f)>4,

• (2) l = 1 and7+k2θ(∞,f)+12δ1+k(0,f)+δ2(0,f)+δ2+k(0,f)+δ(a,f)>k2+5,

• (3) l = 0 and(2k+6)θ(∞,f)+δ2(0,f)+2δ1+k(0,f)+δ2+k(0,f)+θ(0,f)+δ(a,f)>2k+10.

Definition 1.2

Let p₀, p₁, . . . p_k be non-negative integers. We call

a differential monomial in f with degree d_M = p₀ + p₁ + · · · + p_k and weight Γ_M = p₀ + 2p₁ + · · · + (k + 1)p_k, and

where a_j are small functions of f, is called a differential polynomial in f of degree d = max{d_Mj, 1 ≤ j ≤ n} and weight Γ = max{Γ_Mj, 1 ≤ j ≤ n}, furthermore if deg(M_j) = d(j = 1, 2, · · · n), then H[f] is a homogeneous differential polynomial in f of degree d.

In this paper, we improve the above Theorems and obtain the following result.

Theorem 1.3

Let f be a non-constant meromorphic function and H[f] be a non-constant homogeneous differential polynomial of degree d and weight Γ satisfying Γ ≥ (k + 2)d − 2. Let a(z) be a small meromorphic function of f such that a(z) ≢ 0, ∞. Suppose that f − a and H[f] − a share (0, l). Then H[f]-af-a=C for some non-zero constant C if one of the following assumptions holds,

• (i) l ≥ 2 and(1.1)3θ(∞,f)+dδ2+Γ-d(0,fd)+δ2(0,f)+δ(a,f)>4,

• (ii) l = 1 and(1.2)7+Γ-d2θ(∞,f)+d2δ1+Γ-d(0,fd)+dδ2+Γ-d(0,fd)+δ2(0,f)+δ(a,f)>Γ+92,

• (iii) l = 0 and(1.3)[2(Γ-d)+6]θ(∞,f)+δ2(0,f)+dδ2+Γ-d(0,fd)+2dδ1+Γ-d(0,fd)+θ(0,f)+δ(a,f)>2Γ+8.

Especially, when l = 0, i.e. f and H share a IM, if (1.3) holds, then f ≡ H[f].

Lemma 2.1.([11])

Let f be a nonconstant meromorphic function, k be a positive integer. Then

Suppose that F and G are two non-constant meromorphic functions such that F and G share the value 1 IM. Let z₀ be a 1-point of F of order p, a 1-point of G of order q. We denote by NL(r,1F-1) the counting function of those 1-points of F where p > q, by NE1)(r,1F-1) the counting function of those 1-points of F where p = q = 1, by NE(2 the counting function of those 1-points of F where p = q ≥ 2; each point in these counting functions is counted only one time. Similarly, we can define NL(r,1G-1), NE1)(r,1G-1) and NE(2(r,1G-1).

Lemma 2.2.([13])

Let F and G are two nonconstant meromorphic functions,

If F and G share 1 IM and Δ ≢ 0, then

Lemma 2.3

Let H[f] be a non-constant differential polynomial. Let z₀be a pole of f of order p and neither a zero nor a pole of coefficients of H[f]. Then z₀is a pole of H[f] with order at most pd + (Γ − d).

Lemma 2.4

Let f be a transcendental meromorphic function, H[f] is a homogeneous differential polynomial in f of degree d and weight Γ. If H[f] ≢ 0, we have

Lemma 2.5

Let f be a non-constant meromorphic function, H[f] is a homogeneous differential polynomial in f of degree d and weight Γ, and let p be a positive integer. If H[f] ≢ 0 and Γ ≥ (k + 2)d − (p + 1), we have

Let F=H[f]a,G=fa. From the conditions of Theorem 1.3, we know that F and G share (1, l) except the zeros and poles of a(z). From the proof of Lemma 2.4, we have

It is obvious that f is a transcendental meromorphic function. Let Δ be defined by (2.3). We distinguish two cases.

Case 1

Δ ≡ 0. Integrating (2.3), yields

where C and D are constants and C ≠ 0. If there exists a pole z₀ of f with multiplicity p which is not zero or pole of a, then z₀ is a pole of F with multiplicity pd + (Γ − d), a pole of G with multiplicity p. This contradicts (3.2) as H contains at least one derivative. Therefore, we have

(3.2) also shows that F and G share the value 1 CM.

Next, we will prove D = 0.

Suppose D ≠ 0, then we have

So, we have

Subcase 1.1

If CD≠1, then by using (3.3), (3.5) and the second fundamental theorem, we have

This gives that

So we have

Let p = 1, then from assumption we have

Thus from (2.8) in Lemma 2.5, we get

So we have

This gives that

So we have

Since (3.3), we get

Subcase 1.1.1

l ≥ 2.

From δ₂(0, f) + δ(a, f) > 1 and the definition of deficiency, we have

Using the second fundamental theorem of Nevanlinna and (3.3), we have

Combining (3.7) with (3.8), we have

So we have

This gives that

From (3.8), we get a contradiction.

Subcase 1.1.2

l = 1.

When d ≥ 2, by using (1.2) and the definition of deficiency, we get a contradiction.

When d = 1, using the similar method in subcase 1.1.1, we get a contradiction.

Subcase 1.1.3

l = 0.

By using (1.3) and the definition of deficiency, we get a contradiction.

Subcase 1.2

If CD=1, then from (3.4), we have

This gives us that

Using that F=Ha and G=fa, we get

Using (3.3) (3.9), Lemma 2.3 and the first fundamental theorem, we get

which is a contradiction, hence D=0. This gives from (3.2) that

So we get H[f]-af-a=C(C≠0).

Next, we will prove C = 1 when l = 0.

Suppose C ≠ 1, then we have

and

By the second fundamental theorem and (3.3) (3.10), we have

By Lemma 2.5 for p = 1, we have

From the above formula and the definition of deficiency, we have

So we have

Combining (3.11) (3.12) (3.6) with the assumptions of Theorem 1.3, we get a contradiction.

So C = 1 and G ≡ F, i.e. f ≡ H[f].

This is just the conclusion of this theorem.

Case 2

Δ ≢ 0.

By a similar method that used in the proof of Theorem B[16], we get

and

Subcase 2.1

l ≥ 2. It is easy to see that

From (3.13) (3.14) and (3.15), we have

Noting that

Let p = 2, then from assumption we have

Thus, from (2.8) in Lemma 2.5 we obtain that

So we have

This gives that

Which contradicts the assumption (1.1) of Theorem 1.3.

Subcase 2.2

l = 1. Noting that

and

Using the same method as Subcase 2.1, we get

Which contradicts with (1.2) of Theorem 1.3.

Subcase 2.3

l = 0. Noting that

From Lemma 2.5, we have

So we have

Combining (3.13) (3.14) (3.16) with (3.17), we have

From (2.8), we have

Substituting this into (3.18), we have

So we have

Which contradicts the assumption of Theorem 1.3.

Now the proof has been completed.